This invention relates to image reconstruction from tomographic data, and, more particularly, to the definition of discontinuity location and size using limited tomographic data. This invention was made with government support under Contract No. W-7405-ENG-36 awarded by the U.S. Department of Energy. The government has certain rights in the invention.
Tomography produces the reconstruction of a generalized density function f from a large number of line integrals of f. A practically important objective is the reconstruction, from the x-ray data, of functions providing significant information about the physical object, such as the location of discontinuities within an object being interrogated. Tomography equipment for obtaining the attenuation data is well known. For example, in some instances a single source is collimated and traversed across the object, whereby a single sensor output corresponds to a single source transverse location. Here, the source traverses the object at each angular orientation of the object. In other instances, a single source generates a fan-like pattern of radiation that is detected by an array of sensors, where the object is located between the source and the sensor array. The source is then angularly rotated relative to the object to provide a complete set of tomographic data.
Conventional tomography is a global procedure in that the standard convolution formulas for reconstruction of the density at a single point require the line integral data over all lines within some planar cross-section containing the point. Local tomography has been developed for the reconstruction at a point where attenuation data is needed only along lines close to that point within the same cross-section. Local tomography produces the reconstruction of a related function using the square root of the negative Laplace operator and reproduces the locations of discontinuities within an object. See, e.g., E. Vainberg, "Reconstruction of the Internal Three-Dimensional Structure of Objects Based on Real-Time Integral Projections," 17 Sov. J. Nondestr. Test., pp. 415-423 (1981); A. Faridani et al., "Local Tomography," 52 SIAM J. Appl. Math, No. 2, pp. 459-484 (April 1992).
Local tomography can reduce significantly the amount of data needed for the local reconstruction, with a concomitant reduction in x-ray dose. While the location of a discontinuity is reproduced, however, them is no quantitative value for the magnitude of the discontinuity. In many instances it would be useful to know this value in order to make medical, technological, or geophysical conclusions.
Accordingly, it is an object of the present invention to determine both the location and size of discontinuities from tomographic data.
It is another object of the present invention to determine the location and size of discontinuities from only limited attenuation data that includes a region containing the discontinuity.
Additional objects, advantages and novel features of the invention will be set forth in part in the description which follows, and in part will become apparent to those skilled in the art upon examination of the following or may be learned by practice of the invention. The objects and advantages of the invention may be realized and attained by means of the instrumentalities and combinations particularly pointed out in the appended claims.